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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 592504, 10 pages
doi:10.1155/2008/592504
Research Article
Additive Functional Inequalities in
Banach Modules
Choonkil Park,
1
Jong Su An,
2
and Fridoun Moradlou
3
1
Department of Mathematics, Hanyang University, Seoul 133–791, South Korea
2
Department of Mathematics Education, Pusan National University, Pusan 609–735, South Korea
3
Faculty of Mathematical Science, University of Tabriz, Tabriz 5166 15731, Iran
Correspondence should be addressed to Jong Su An,
Received 1 April 2008; Revised 4 June 2008; Accepted 10 November 2008
Recommended by Alberto Cabada
We investigate the following functional inequality 2fx2fy2fz − fx y − fy z≤
fx z in Banach modules over a C
∗
-algebra and prove the generalized Hyers-Ulam stability
of linear mappings in Banach modules over a C
∗
-algebra in the spirit of the Th. M. Rassias stability
approach. Moreover, these results are applied to investigate homomorphisms in complex Banach
algebras and prove the generalized Hyers-Ulam stability of homomorphisms in complex Banach
algebras.
Copyright q 2008 Choonkil Park et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introduction and preliminaries
The stability problem of functional equations originated from a question of Ulam 1
concerning the stability of group homomorphisms. Hyers 2 gave a first affirmative partial
answer to the question of Ulam for Banach spaces. The Hyers theorem was generalized by
Aoki 3 for additive mappings and by Th. M. Rassias 4 for linear mappings by considering
an unbounded Cauchy difference. The paper of Th. M. Rassias 4 has provided a lot of
influence in the development of what we call generalized Hyers-Ulam stability of functional
equations. A generalization of the Th. M. Rassias theorem was obtained by G
˘
avrut¸a 5 by
replacing the unbounded Cauchy difference by a general control function in the spirit of
the Th. M. Rassias approach. Th. M. Rassias 6 during the 27th International Symposium
on Functional Equations asked the question whether such a theorem can also be proved for
p ≥ 1. Gajda 7, following the same approach as in Th. M. Rassias 4, gave an affirmative
solution to this question for p>1. It was shown by Gajda 7 as well as by Th. M. Rassias
and
ˇ
Semrl 8 that one cannot prove a Th. M. Rassias-type theorem when p 1. J. M. Rassias
9 followed the innovative approach of the Th. M. Rassias theorem in which he replaced the
factor x
p
y
p
by x
p
·y
q
for p, q ∈ R with p q
/
1. During the last three decades, a
number of papers and research monographs have been published on various generalizations
2 Journal of Inequalities and Applications
and applications of the generalized Hyers-Ulam stability to a number of functional equations
and mappings see 10–18.
Gil
´
anyi 19 showed that if f satisfies the functional inequality
2fx2fy − fx − y
≤
fx y
, 1.1
then f satisfies the Jordan-von Neumann functional equation
2fx2fyfx yfx − y. 1.2
See also 20. Fechner 21 and Gil
´
anyi 22 proved the generalized Hyers-Ulam stability of
the functional inequality
1.1.
In this paper, we investigate an A-linear mapping associated with the functional
inequality
2fx2fy2fz − fx y − fy z
≤
fx z
1.3
and prove the generalized Hyers-Ulam stability of A-linear mappings in Banach A-modules
associated with the functional inequality 1.3. These results are applied to investigate
homomorphisms in complex Banach algebras and prove the generalized Hyers-Ulam
stability of homomorphisms in complex Banach algebras.
2. Functional inequalities in Banach modules over a C
∗
-algebra
Throughout this section, let A be a unital C
∗
-algebra with unitary group UA and unit e and
B aunitalC
∗
-algebra. Assume that X is a Banach A-module with norm ·
X
and that Y is a
Banach A-module with norm ·
Y
.
Lemma 2.1. Let f : X → Y be a mapping such that
2ufx2fy2fz − fux y − fy z
Y
≤
fux z
Y
2.1
for all x, y, z ∈ X and all u ∈ UA.Thenf is A-linear.
Proof. Letting x y z 0andu e ∈ UA in 2.1,weget
4f0
Y
≤
f0
Y
. 2.2
So f00.
Letting u e ∈ UA, y 0andz −x in 2.1,weget
fxf−x
Y
≤
f0
Y
0 2.3
for all x ∈ X. Hence f−x−fx for all x ∈ X.
Choonkil Park et al. 3
Letting z −x and u e ∈ UA in 2.1,weget
2fx2fy2f−x − fx y − fy − x
Y
2fy − fy x − fy − x
Y
≤
f0
Y
0
2.4
for all x, y ∈ X.Sofy xfy − x2fy for all x, y ∈ X.Thus
fx yfxfy2.5
for all x, y ∈ X.
Letting z −ux and y 0in2.1,weget
2ufx − 2fux
Y
2ufx2f−uz
Y
≤
f0
Y
0
2.6
for all x ∈ X and all u ∈ UA.Thus
fuzufz2.7
for all u ∈ UA and all z ∈ X.Now,leta ∈ Aa
/
0 and M an integer greater than 4|a|. Then
|a/M| < 1/4 < 1 − 2/3 1/3. By 23, Theorem 1, there exist three elements u
1
,u
2
,u
3
∈ UA
such that 3a/Mu
1
u
2
u
3
.Soby2.7
faxf
M
3
·3
a
M
x
M·f
1
3
·3
a
M
x
M
3
f
3
a
M
x
M
3
f
u
1
x u
2
x u
3
x
M
3
f
u
1
x
f
u
2
x
f
u
3
x
M
3
u
1
u
2
u
3
fx
M
3
·3
a
M
fx
afx
2.8
for all x ∈ X.Sof : X → Y is A-linear, as desired.
4 Journal of Inequalities and Applications
Now, we prove the generalized Hyers-Ulam stability of A-linear mappings in Banach
A-modules.
Theorem 2.2. Let r>1 and θ be nonnegative real numbers, and let f : X → Y be an odd mapping
such that
2ufx2fy2fz − fux y − fy z
Y
≤
fux z
Y
θ
x
r
X
y
r
X
z
r
X
2.9
for all x, y, z ∈ X and all u ∈ UA. Then there exists a unique A-linear mapping L : X → Y such
that
fx − Lx
Y
≤
3θ
2
r
− 2
x
r
X
2.10
for all x ∈ X.
Proof. Since f is an odd mapping, f−x−fx for all x ∈ X.Sof00.
Letting u e ∈ UA, y x and z −x in 2.9,weget
2fx − f2x
Y
2fxf−2x
Y
≤ 3θx
r
X
2.11
for all x ∈ X.So
fx − 2f
x
2
Y
≤
3
2
r
θx
r
X
2.12
for all x ∈ X. Hence
2
l
f
x
2
l
− 2
m
f
x
2
m
Y
≤
m−1
jl
2
j
f
x
2
j
− 2
j1
f
x
2
j1
Y
≤
3
2
r
m−1
jl
2
j
2
rj
θx
r
X
2.13
for all nonnegative integers m and l with m>land all x ∈ X. It follows from 2.13 that the
sequence {2
n
fx/2
n
} is Cauchy for all x ∈ X. Since Y is complete, the sequence {2
n
fx/2
n
}
converges. So one can define the mapping L : X → Y by
Lx : lim
n →∞
2
n
f
x
2
n
2.14
for all x ∈ X. Moreover, letting l 0 and passing the limit m →∞in 2.13,weget2.10.
Choonkil Park et al. 5
It follows from 2.9 that
2uLx2Ly2Lz − Lux y − Ly z
Y
lim
n →∞
2
n
2uf
x
2
n
2f
y
2
n
2f
z
2
n
− f
ux y
2
n
− f
y z
2
n
≤ lim
n →∞
2
n
f
ux z
2
n
Y
lim
n →∞
2
n
θ
2
nr
x
r
X
y
r
X
z
r
X
Lux z
Y
2.15
for all x, y, z ∈ X and all u ∈ UA.So
2uLx2Ly2Lz − Lux y − Ly z
Y
≤
Lux z
Y
2.16
for all x, y, z ∈ X and all u ∈ UA.ByLemma 2.1, the mapping L : X → Y is A-linear.
Now, let T : X → Y be another A-linear mapping satisfying 2.10. Then, we have
Lx − T x
Y
2
n
L
x
2
n
− T
x
2
n
Y
≤ 2
n
L
x
2
n
− f
x
2
n
Y
T
x
2
n
− f
x
2
n
Y
≤
6·2
n
2
r
− 2
2
nr
θx
r
X
,
2.17
which tends to zero as n →∞for all x ∈ X. So we can conclude that LxTx for all
x ∈ X. This proves the uniqueness of L. Thus the mapping L : X → Y is a unique A-linear
mapping satisfying 2.10.
Theorem 2.3. Let r<1 and θ be positive real numbers, and let f : X → Y be an odd mapping
satisfying 2.9. Then there exists a unique A-linear mapping L : X → Y such that
fx − Lx
Y
≤
3θ
2 − 2
r
x
r
X
2.18
for all x ∈ X.
Proof. It follows from 2.11 that
fx −
1
2
f2x
Y
≤
3
2
θx
r
X
2.19
6 Journal of Inequalities and Applications
for all x ∈ X. Hence
1
2
l
f
2
l
x
−
1
2
m
f
2
m
x
Y
1
2
l
f
2
l
x
−
1
2
m
f
2
m
x
Y
≤
3
2
m−1
jl
2
rj
2
j
θx
r
X
2.20
for all nonnegative integers m and l with m>land all x ∈ X. It follows from 2.20 that
the sequence {1/2
n
f2
n
x} is Cauchy for all x ∈ X. Since Y is complete, the sequence
{1/2
n
f2
n
x} converges. So one can define the mapping L : X → Y by
Lx : lim
n →∞
1
2
n
f
2
n
x
2.21
for all x ∈ X. Moreover, letting l 0 and passing the limit m →∞in 2.20,weget2.18.
The rest of the proof is similar to the proof of Theorem 2.2.
Theorem 2.4. Let r>1/3 and θ be nonnegative real numbers, and let f : X → Y be an odd
mapping such that
2ufx2fy2fz − fux y − fy z
Y
≤
fux z
Y
θ·x
r
X
·y
r
X
·z
r
X
2.22
for all x, y, z ∈ X and all u ∈ UA. Then there exists a unique A-linear mapping L : X → Y such
that
fx − Lx
Y
≤
θ
8
r
− 2
x
3r
X
2.23
for all x ∈ X.
Proof. Since f is an odd mapping, f−x−fx for all x ∈ X.Sof00.
Letting u e ∈ UA, y x, and z −x in 2.22,weget
2fx − f2x
Y
2fxf−2x
Y
≤ θx
3r
X
2.24
for all x ∈ X.So
fx − 2f
x
2
Y
≤
θ
8
r
x
3r
X
2.25
Choonkil Park et al. 7
for all x ∈ X. Hence
2
l
f
x
2
l
− 2
m
f
x
2
m
Y
≤
m−1
jl
2
j
f
x
2
j
− 2
j1
f
x
2
j1
Y
≤
θ
8
r
m−1
jl
2
j
8
rj
x
3r
X
2.26
for all nonnegative integers m and l with m>land all x ∈ X. It follows from 2.26 that the
sequence {2
n
fx/2
n
} is Cauchy for all x ∈ X. Since Y is complete, the sequence {2
n
fx/2
n
}
converges. So one can define the mapping L : X → Y by
Lx : lim
n →∞
2
n
f
x
2
n
2.27
for all x ∈ X. Moreover, letting l 0 and passing the limit m →∞in 2.26,weget2.23.
The rest of the proof is similar to the proof of Theorem 2.2.
Theorem 2.5. Let r<1/3 and θ be positive real numbers, and let f : X → Y be an odd mapping
satisfying 2.22. Then there exists a unique A-linear mapping L : X → Y such that
fx − Lx
Y
≤
θ
2 − 8
r
x
3r
X
2.28
for all x ∈ X.
Proof. It follows from 2.24 that
fx −
1
2
f2x
Y
≤
θ
2
x
3r
X
2.29
for all x ∈ X. Hence
1
2
l
f
2
l
x
−
1
2
m
f
2
m
x
Y
≤
m−1
jl
1
2
j
f
2
j
x
−
1
2
j1
f
2
j1
x
Y
≤
θ
2
m−1
jl
8
rj
2
j
x
3r
X
2.30
for all nonnegative integers m and l with m>land all x ∈ X. It follows from 2.30 that
the sequence {1/2
n
f2
n
x} is Cauchy for all x ∈ X. Since Y is complete, the sequence
8 Journal of Inequalities and Applications
{1/2
n
f2
n
x} converges. So one can define the mapping L : X → Y by
Lx : lim
n →∞
1
2
n
f
2
n
x
2.31
for all x ∈ X. Moreover, letting l 0 and passing the limit m →∞in 2.30,weget2.28.
The rest of the proof is similar to the proof of Theorem 2.2.
3. Generalized Hyers-Ulam stability of homomorphisms in Banach algebras
Throughout this section, let A and B be complex Banach algebras.
Proposition 3.1. Let f : A → B be a multiplicative mapping such that
2μfx2fy2fz − fμx y − fy z
≤
fμx z
3.1
for all x, y, z ∈ A and all μ ∈ T : {λ ∈ C ||λ| 1}.Thenf is an algebra homomorphism.
Proof. Every complex Banach algebra can be considered as a Banach module over C.By
Lemma 2.1, the mapping f : A → B is a C-linear. So the multiplicative mapping f : A
→ B
is an algebra homomorphism.
Now, we prove the generalized Hyers-Ulam stability of homomorphisms in complex
Banach algebras.
Theorem 3.2. Let r>1 and θ be nonnegative real numbers, and let f : A → B be an odd
multiplicative mapping such that
2μfx2fy2fz − fμx y − fy z
≤
fμx z
θ
x
r
y
r
z
r
3.2
for all x, y, z ∈ A and all μ ∈ T. Then there exists a unique algebra homomorphism H : A → B such
that
fx − Hx
≤
3θ
2
r
− 2
x
r
3.3
for all x ∈ A.
Proof. By Theorem 2.2, there exists a unique C-linear mapping H : A → B satisfying 3.3.
The mapping H : A → B is given by
Hx : lim
n →∞
2
n
f
x
2
n
3.4
for all x ∈ A.
Choonkil Park et al. 9
Since f : A → B is multiplicative,
Hxy lim
n →∞
4
n
f
xy
4
n
lim
n →∞
2
n
f
x
2
n
·2
n
f
y
2
n
HxHy
3.5
for all x, y ∈ A. Thus t he mapping H : A → B is an algebra homomorphism satisfying
3.3.
Theorem 3.3. Let r<1 and θ be positive real numbers, and let f : A → B be an odd multiplicative
mapping satisfying 3.2. Then there exists a unique algebra homomorphism H : A → B such that
fx − Hx
≤
3θ
2 − 2
r
x
r
3.6
for all x ∈ A.
Proof. The proof is similar to the proofs of Theorems 2.3 and 3.2.
Theorem 3.4. Let r>1/3 and θ be nonnegative real numbers, and let f : A → B be an odd
multiplicative mapping such that
2μfx2fy2fz − fμx y − fy z
≤
fμx z
θ·x
r
·y
r
·z
r
3.7
for all x, y, z ∈ A and all μ ∈ T. Then there exists a unique algebra homomorphism H : A → B such
that
fx − Hx
≤
θ
8
r
− 2
x
3r
3.8
for all x ∈ A.
Proof. The proof is similar to the proofs of Theorems 2.4 and 3.2.
Theorem 3.5. Let r<1/3 and θ be positive real numbers, and let f : A → B be an odd
multiplicative mapping satisfying 3.7. Then there exists a unique algebra homomorphism H : A →
B such that
fx − Hx
≤
θ
2 − 8
r
x
3r
3.9
for all x ∈ A.
Proof. The proof is similar to the proofs of Theorems 2.5 and 3.2.
10 Journal of Inequalities and Applications
Acknowledgments
The first author was supported by Korea Research Foundation Grant KRF-2007-313-C00033
and the authors would like to thank the referees for a number of valuable suggestions
regarding a previous version of this paper.
References
1 S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied
Mathematics, no. 8, Interscience, New York, NY, USA, 1960.
2 D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of
Sciences of the United States of America, vol. 27, no. 4, pp. 222–224, 1941.
3 T. Aoki, “On the stability of the linear transformation in Banach spaces,” Journal of the Mathematical
Society of Japan, vol. 2, pp. 64–66, 1950.
4 Th. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American
Mathematical Society, vol. 72, no. 2, pp. 297–300, 1978.
5 P. G
˘
avrut¸a, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive
mappings,” Journal of Mathematical Analysis and Applications, vol. 184, no. 3, pp. 431–436, 1994.
6 Th. M. Rassias, “Problem 16; 2, report of the 27th International Symposium on Functional Equations,”
Aequationes Mathematicae, vol. 39, no. 2-3, pp. 292–293, 309, 1990.
7 Z. Gajda, “On stability of additive mappings,” International Journal of Mathematics and Mathematical
Sciences, vol. 14, no. 3, pp. 431–434, 1991.
8 Th. M. Rassias and P.
ˇ
Semrl, “On the behavior of mappings which do not satisfy Hyers-Ulam
stability,” Proceedings of the American Mathematical Society, vol. 114, no. 4, pp. 989–993, 1992.
9 J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,” Journal of
Functional Analysis, vol. 46, no. 1, pp. 126–130, 1982.
10 S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ,
USA, 2002.
11 D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in S everal Variables, vol. 34 of
Progress in Nonlinear Differential Equations and Their Applications,Birkh
¨
auser, Boston, Mass, USA, 1998.
12 K W. Jun and Y H. Lee, “A generalization of the Hyers-Ulam-Rassias stability of the Pexiderized
quadratic equations,” Journal of Mathematical Analysis and Applications, vol. 297, no. 1, pp. 70–86, 2004.
13 S M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis,Hadronic
Press, Palm Harbor, Fla, USA, 2001.
14 C G. Park, “On the stability of the linear mapping in Banach modules,” Journal of Mathematical
Analysis and Applications, vol. 275, no. 2, pp. 711–720, 2002.
15 C G. Park, “Modified Trif’s functional equations in Banach modules over a C
∗
-algebra and
approximate algebra homomorphisms,” Journal of Mathematical Analysis and Applications, vol. 278, no.
1, pp. 93–108, 2003.
16 C G. Park, “On an approximate automorphism on a C
∗
-algebra,” Proceedings of the American
Mathematical Society, vol. 132, no. 6, pp. 1739–1745, 2004.
17 C G. Park, “Lie ∗-homomorphisms between Lie C
∗
-algebras and Lie ∗-derivations on Lie C
∗
-
algebras,” Journal of Mathematical Analysis and Applications, vol. 293, no. 2, pp. 419–434, 2004.
18 C G. Park, “Homomorphisms between Poisson JC
∗
-algebras,” Bulletin of the Brazilian Mathematical
Society, vol. 36, no. 1, pp. 79–97, 2005.
19 A. Gil
´
anyi, “Eine zur Parallelogrammgleichung
¨
aquivalente Ungleichung,” Aequationes Mathematicae,
vol. 62, no. 3, pp. 303–309, 2001.
20 J. R
¨
atz, “On inequalities associated with the Jordan-von Neumann functional equation,” Aequationes
Mathematicae, vol. 66, no. 1-2, pp. 191–200, 2003.
21 W. Fechner, “Stability of a functional inequality associated with the Jordan-von Neumann functional
equation,” Aequationes Mathematicae, vol. 71, no. 1-2, pp. 149–161, 2006.
22 A. Gil
´
anyi, “On a problem by K. Nikodem,” Mathematical Inequalities & Applications, vol. 5, no. 4, pp.
707–710, 2002.
23 R. V. Kadison and G. K. Pedersen, “Means and convex combinations of unitary operators,”
Mathematica Scandinavica, vol. 57, no. 2, pp. 249–266, 1985.
